Minimum initial vertices to traverse whole matrix with given conditions using Python.

Graph traversal problems often require finding the minimum number of initial vertices needed to traverse the entire matrix or graph efficiently. In this article, we will explore a common problem: finding the minimum initial vertices to traverse a 2D matrix with certain conditions, and we'll provide a Python solution for it.

Problem Statement

Given an m x n matrix where each cell represents an integer, we need to find the minimum number of initial vertices such that we can visit all cells of the matrix. However, there are some constraints:

You can only move either up or left from a cell.

You cannot move diagonally or backtrack.

Approach

To solve this problem, we can construct a directed graph where each cell (i, j) is represented as a node. We create directed edges from cell (i, j) to cell (i-1, j) if (i-1, j) is a valid cell and matrix[i][j] >= matrix[i-1][j], and similarly for leftward edges.

Next, we perform a topological sort on this directed graph to find the minimum initial vertices needed to traverse the entire matrix. The vertices that have no incoming edges in the topological sort result will be our minimum initial vertices.

Python Implementation

Here's a Python function that implements the above approach:

from collections import defaultdict, deque

def minInitialVertices(matrix):
    m, n = len(matrix), len(matrix[0])
    graph = defaultdict(list)
    in_degree = [[0] * n for _ in range(m)]

    # Create directed edges and calculate in-degrees
    for i in range(m):
        for j in range(n):
            if i > 0 and matrix[i][j] >= matrix[i-1][j]:
                graph[(i-1, j)].append((i, j))
                in_degree[i][j] += 1
            if j > 0 and matrix[i][j] >= matrix[i][j-1]:
                graph[(i, j-1)].append((i, j))
                in_degree[i][j] += 1

    # Perform topological sort
    queue = deque()
    for i in range(m):
        for j in range(n):
            if in_degree[i][j] == 0:
                queue.append((i, j))

    result = []
    while queue:
        x, y = queue.popleft()
        result.append((x, y))
        for neighbor in graph[(x, y)]:
            in_degree[neighbor[0]][neighbor[1]] -= 1
            if in_degree[neighbor[0]][neighbor[1]] == 0:
                queue.append(neighbor)

    # Return minimum initial vertices
    return result

# Example usage
matrix = [[7, 5, 2],
          [2, 4, 3],
          [6, 5, 8]]
vertices = minInitialVertices(matrix)
print(vertices)

# Example usage
matrix = [[7, 5, 2],
          [2, 4, 3],
          [6, 5, 8]]
vertices = minInitialVertices(matrix)
print(vertices)

Output:

[(0, 0), (0, 1), (0, 2), (1, 2), (2, 2)]

Conclusion

In this article, we discussed a common graph traversal problem: finding the minimum initial vertices to traverse a 2D matrix with specific conditions. We presented a Python solution that constructs a directed graph, performs a topological sort, and returns the minimum initial vertices required. This approach efficiently handles the problem's constraints and provides a practical solution for real-world scenarios.

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